Empirical Analyses of Income: Finland (2009) and Australia (1967-1968)

Analyses of income data are often based on assumptions concerning theoretical distributions. In this study, we apply statistical analyses, but ignore specific distribution models. The main income data sets considered in this study are taxable income in Finland (2009) and household income in Australia (1967-1968). Our intention is to compare statistical analyses performed without assumptions of the theoretical models with earlier results based on specific models. We have presented the central objects, probability density function, cumulative distribution function, the Lorenz curve, the derivative of the Lorenz curve, the Gini index and the Pietra index. The trapezium rule, Simpson´s rule, the regression model and the difference quotients yield comparable results for the Finnish data, but for the Australian data the differences are marked. For the Australian data, the discrepancies are caused by limited data. JEL classification numbers: D31, D63, E64. Keywords: Cumulative distribution function, Probability density function, Mean, quantiles, Lorenz curve, Gini coefficient, Pietra index, Robin Hood index, Trapezium rule, Simpson´s rule, Regression models, Difference quotients, Derivative of Lorenz curve

On Taylor Series Expansion for Statistical Moments of Functions of Correlated Random Variables

The paper is focused on Taylor series expansion for statistical analysis of functions of random variables with special attention to correlated input random variables. It is shown that the standard approach leads to significant deviations in estimated variance of non-linear functions. Moreover, input random variables are often correlated in industrial applications; thus, it is crucial to obtain accurate estimations of partial derivatives by a numerical differencing scheme. Therefore, a novel methodology for construction of Taylor series expansion of increasing complexity of differencing schemes is proposed and applied on several analytical examples. The methodology is adapted for engineering applications by proposed asymmetric difference quotients in combination with a specific step-size parameter. It is shown that proposed differencing schemes are suitable for functions of correlated random variables. Finally, the accuracy, efficiency, and limitations of the proposed methodology are discussed.

Analysis of thermal flow in waterwall tubes of the combustion chamber depending on the fluid parameters

The article presents the method of determining the temperature distribution in waterwall tubes of the combustion chamber. To simulate the operating conditions of waterwall tubes have been selected the model with distributed parameters, which is based on the solution of equations of the energy, mass and momentum conservation laws. The purpose of the calculations is determining the enthalpy, mass-flow and pressure of the working fluid flowing inside the tubes. The balance equations have been transformed into a form in which spatial derivatives are on the left, and the right side contains time derivatives. Then the time derivatives were replaced with backward difference quotients, and the obtained system of differential equations was solved by the Runge-Kutta method. The analysis takes into account the variability of fluid parameters depending on the mass-flow at the inlet of the tube and heat flux on the surface of the tube. The analysis of fluid parameters was carried out based on operating parameters occurring in one of the Polish supercritical power plants. Then it was compared with characteristics for systems operating at increased or reduced thermal flux on the walls of the furnace or systems operating at increased or reduced mass-flow of the working fluid at the inlet to the waterwall tube.

Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients

We establish a consistency result by comparing two independent notions of generalized solutions to a large class of linear hyperbolic first-order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of [Formula: see text]-solutions which we recently introduced and arose in connection to the vectorial calculus of variations in [Formula: see text] and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.